1. Field of the Invention
The present invention relates to systems and methods for modeling, simulating and analyzing the motion of two fluids and the interface between those fluids using the level set method.
2. Description of the Related Art
An ink-jet print head is a printing device which produces images by ejecting ink droplets onto a print medium. Control of the ink ejection process and the ensuing ink droplet is essential to ensuring the quality of any product created by the print head. To achieve such control it is important to have accurate and efficient simulations of the printing and ejection process. Simulating this process includes modeling of at least two fluids (i.e., ink and air) and the interface between these fluids. Prior art methods have used computational fluid dynamics, finite element analysis, finite difference analysis, and level set methods to model this behavior.
The level set method though useful sometimes has problems with mass conservation. Mass conservation refers to the ability of a simulation method to maintain, for a closed system, the mass of the simulated system. Similarly, for an open system, any masses added or subtracted should be reflected in the simulated mass of the system. Prior art methods have addressed the issue of mass conservation by periodically re-distancing the level set or using finer meshes for the entire simulation.
While using finer meshes does improve the mass conservation of the simulation, it also leads to a tremendous increase in the time and the resources required of the simulation. For example, if we started with an n×m mesh for a two-dimensional simulation. We would have to store 2×n×m velocities, n×m pressures, and n×m level sets. To enforce the fluid incompressibility, we have to solve a linear system with an n×m by n×m coefficient matrix. If we use a twice finer mesh for all the variables, we have to store 2×2n×2m velocities, 2n×2m pressures, and 2n×2m level sets. We also have to solve a linear system with a 2n×2m by 2n×2m coefficient matrix. Thus, the storage requirements are quadrupled if the mesh is twice finer. In addition, we will have to solve a linear system with a coefficient matrix that contains four times as many nonzero elements. If a direct solver is used, a four times more populated coefficient matrix usually requires computation time that is 64 times longer. Furthermore, if an explicit time integration scheme is used then a time step that is four times smaller would be required. This would further increase the required computation time.
Therefore, what is needed is a simulation method that improves upon the mass conservation of the level set method without unduly burdening computational resources.